Kelly was born in Toronto and grew up one hour east of the city in Bowmanville. He started playing in the Clarington Recreational Hockey League and with the Clarington Toros AA program. For one year, he played for the Toronto Marlboros bantam team and then OHA Jr. A. hockey with the Aurora Tigers. Kelly was then selected in the fourth round, 56th overall, by the Ontario Hockey League (OHL)'s London Knights in the 1997 OHL Priority Selection.

At the major junior level, Kelly played for both the Knights and the Sudbury Wolves. In 1998–99, he scored 36 goals, his career-high scoring mark in the OHL. He also played for the Team Orr in the mid-season at the CHL Top Prospects Game. In the OHL playoffs, he scored nine goals and 26 points in 25 games as the Knights reached the OHL Final.

Kelly made his NHL debut on February 5, 2004, in a match against the Toronto Maple Leafs, one of four games he appeared in with Ottawa during 2003–04. In his rookie season of 2005–06, he became a regular in Ottawa's line-up, appearing in all of the team's 82 games and registering 30 points whilst playing in a checking role. The following season, he was a member of the Senators team that advanced to the 2007 Stanley Cup Finals.[3]

On July 31, 2007, he re-signed with the Senators to a one-year contract worth $1.263 million. Kelly was set to become an unrestricted free agent on July 1, 2008, but again re-signed with the Senators on a four-year contract extension worth $8.5 million on June 20, 2008.

As part of a rebuilding process undertaken by the Senators as the 2010–11 season was concluding, Kelly was dealt to the Boston Bruins on February 15, 2011, for a second-round draft pick in 2011; Ottawa later used the pick to select forward Shane Prince. On June 15, 2011, Kelly and the Boston Bruins won the Stanley Cup in seven games over the Vancouver Canucks.

A pending unrestricted free agent as the 2011–12 season wrapped up, Kelly was rumoured to be returning to the Senators,[4] though he ended up re-signing with the Bruins on June 11, 2012, agreeing to a four-year, $12 million contract.[5] The deal, however, was quickly rejected by the NHL due to what then-Bruins General Manager Peter Chiarelli referred to as "payroll tagging issues."[6]

As the 2013–14 season began on October 3, 2013, with a home game against the Tampa Bay Lightning, Kelly had his first-ever chance at a penalty shot in his NHL career in the first period of the game; he converted the penalty shot, scoring the first Bruins goal of the season against goaltender Anders Lindbäck while the Bruins were in a short-handed situation en route to an eventual 3–1 home victory.[7] The goal marked the first time in NHL history that a team scored its first goal of the season via a penalty shot.[8]

In his sixth year with the Bruins in the 2015–16 season, and in the final year of the his contract, Kelly began the campaign leading the Bruins penalty-kill. Kelly scored 2 goals in 11 games before on November 3, 2015, he buckled his left leg on the ice and broke his left femur in a game against the Dallas Stars.[9] He was announced to have undergone surgery the following day and was scheduled to have a 6-8 month recovery period, which effectively ruled him out for the season.[10]

As a free agent following his recovery from his broken leg with the Bruins, Kelly signed a one-year contract to return to the Ottawa Senators on July 7, 2016.[11] In the 2016–17 season, Kelly appeared in all 82 games for the club, scoring 12 points.

At the conclusion of his contract, Kelly was not re-signed by the Senators, making him an unrestricted free agent. Unsigned over the summer, the Edmonton Oilers signed Kelly to a professional tryout on September 9, 2017.[12] Kelly attended the Oilers training camp and pre-season and remained with the club to start the 2017–18 season. Despite practicing with the team, Kelly was not offered a contract with the Oilers and on November 24, 2017, he returned within the Senators organization by signing a professional tryout deal with new AHL affiliate, the Belleville Senators.[13]

On December 20, 2017, Kelly was released from his professional tryout agreement with the Belleville Senators to join Team Canada for the 2017 Spengler Cup,[16] which they won. He rejoined the Senators immediately following the tournament.

Kelly married during the summer of 2008 in Mexico.[19] He met his wife Krissy Broderick while attending Saunders Secondary School as a member of the London Knights as a teenager. Broderick now teaches elementary school in Ottawa.[20]